Set up the system of equations and then solve it by using an inverse matrix.
One safe investment pays 10% per year, and a more risky investment pays 18% per year. A woman has $144,800 to invest and would like to have an income of $20,000 per year from her investments. How much should she invest at each rate?
10% rate $
18% rate $
To set up the system of equations, we can use the following information:
Let x represent the amount of money invested at the 10% rate.
Let y represent the amount of money invested at the 18% rate.
We can form the equations based on the given information:
Equation 1: The total amount invested should be $144,800.
x + y = 144800
Equation 2: The income generated from the investments should be $20,000 per year.
0.10x + 0.18y = 20000
To solve this system of equations using an inverse matrix, we can represent the system in matrix form:
| 1 1 | | x | | 144800 |
| 0.10 0.18 | * | y | = | 20000 |
We need to find the inverse of the coefficient matrix:
| 1 1 |
| 0.10 0.18 |
Calculating the inverse matrix will give us:
| 0.6 -1.67 |
| -0.1 8.33 |
Now, multiplying the inverse of the coefficient matrix with the constant matrix gives us the values of x and y:
| 0.6 -1.67 | | 144800 | | x |
| -0.1 8.33 | * | 20000 | = | y |
Simplifying the matrix equation, we have:
0.6x - 1.67y = 144800
-0.1x + 8.33y = 20000
We can solve this system of equations to find the values of x and y.
If she invests x an y, then we have
x+y = 144800
.10x + .18y = 20000
Now just do that as a matrix equation.